Jones was dismissed after just 18 minutes for dissent and Leeds were 3-0 up before the break with goals from Neil Kilkenny, Luciano Becchio and Robert Snodgrass. Tom Pope netted twice as the Railwaymen rallied in the second half but the visitors held on.
Liam Dickinson twice tested John Ruddy in the Crewe goal before Jones' straight red gave the Railwaymen a mountain to climb.
It came as no surprise when Kilkenny drilled home from the edge of the box to put the visitors in front in the 27th minute. And two minutes later Snodgrass crossed for Becchio to double the lead.
Crewe were all at sea and Snodgrass made no mistake when Steven Schumacher gave the ball away in the 32nd minute, racing clear and sticking the ball under Ruddy at the near post.
The Alex didn't give up and Pope pulled one back, heading in Gylfi Sigurdsson's 60th-minute corner. An unlikely comeback suddenly looked on when Pope stooped to nod in Danny Woodards's deflected shot with 18 minutes to go but United held on.